Group1- Freny Mehta

Laws in Probability:
1. Addition law-

The addition rule is a result used to determine the probability that event A or event B occurs or both occur.

The result is often written as follows, using set notation:

where:

P(A) = probability that event A occurs

P(B) = probability that event B occurs

= probability that event A or event B occurs

= probability that event A and event B both occur

For mutually exclusive events, that is events which cannot occur together:

= 0

The addition rule therefore reduces to

= P(A) + P(B)

For independent events, that is events which have no influence on each other:

The addition rule therefore reduces to

Example

Suppose we wish to find the probability of drawing either a king or a spade in a single draw from a pack of 52 playing cards.

We define the events A = 'draw a king' and B = 'draw a spade'

Since there are 4 kings in the pack and 13 spades, but 1 card is both a king and a spade, we have:

= 4/52 + 13/52 - 1/52 = 16/52

So, the probability of drawing either a king or a spade is 16/52 (= 4/13). 

 

2. Conditional law- It  is the probability that an event will occur, when another event is known to occur or to have occurred. 

For Example:

Suppose that somebody secretly rolls two fair six-sided dice, and we must predict the outcome.

• Let A be the value rolled on die 1
• Let B be the value rolled on die 2

What is the probability that A = 2? Table 1 shows the sample space. A = 2 in 6 of the 36 outcomes, thus P(A=2) = ⁶⁄₃₆ = ¹⁄₆.

Table

+	B=1	2	3	4	5	6
A=1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

 

Suppose it is revealed that A+B ≤ 5. Table 2 shows that A+B ≤ 5 for 10 outcomes. For 3 of these, A = 2. So the probability that A = 2 given that A+B ≤ 5 is P(A=2 | A+B ≤ 5) = ³⁄₁₀ = 0.3.

Table 2

+	B=1	2	3	4	5	6
A=1	2	3	4	5	6	7
2	3	4	5	6	7	8
3	4	5	6	7	8	9
4	5	6	7	8	9	10
5	6	7	8	9	10	11
6	7	8	9	10	11	12

3. Multiplicative law-It is used for dependent events.

For Example : Suppose you are going to draw two cards from a standard deck. What is the probability that the first card is an ace and the second card is a jack (just one of several ways to get “blackjack” or 21).

 

Using the multiplication rule we get

 

P(ace) P(jack) = (4/52)(4/51) = 16/2652 = 4/663

Sources: 
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_ProbabilityMultiplicationRule.xml.
http://www.stat.wvu.edu/SRS/Modules/ProbLaw/AndProb.html.
http://www.mathgoodies.com/lessons/vol6/addition_rules.html.
http://www3.ul.ie/~mlc/support/Loughborough%20website/chap4/4_3.pdf.